2.2.7 · D5Fluid Mechanics
Question bank — Buoyancy — Archimedes' principle, derivation from pressure difference
Before we start, three words we lean on constantly:
- Displaced volume ::: The volume of fluid the object shoves out of the way — equals the object's submerged volume, not always its total volume.
- Apparent weight ::: What a scale reads while the object hangs in fluid: . See Apparent weight and weighing methods.
- Average density ::: Total mass ÷ total outside volume (including any enclosed air). This is what decides floating, not the material's own density.
True or false — justify
A block twice as deep in the same water feels twice the buoyant force.
False — buoyancy comes from the pressure difference , which depends on the block's height/volume, not its absolute depth. See Pressure in fluids — hydrostatic pressure.
Two objects of equal volume, fully submerged in the same fluid, feel equal buoyant force even if one is lead and one is wood.
True — contains only fluid density and displaced volume; the object's own density never appears in it.
A floating object always displaces its own full volume of fluid.
False — a floating object displaces only enough fluid to match its weight; only the submerged fraction of its volume is under the surface.
The buoyant force on a fully submerged object equals its weight.
False — that equality holds only at floating equilibrium; a fully submerged sinking object has , and a submerged held-down cork has .
Atmospheric pressure adds a downward push that reduces buoyancy.
False — atmosphere () presses equally on top and bottom faces, so it cancels in and never affects .
A heavier object always sinks faster/more surely than a lighter one.
False — sinking is decided by density, not weight. A 300-tonne steel ship floats while a 1-gram steel ball sinks, because the ship's average density is lower.
If you lower the water level so the object is only half submerged, its buoyant force halves.
True — displaced volume halves, so halves; this is exactly how a floating body finds its equilibrium depth.
An air bubble rising in water experiences an upward buoyant force.
True — the water is far denser than the bubble's air, so hugely exceeds the bubble's tiny weight; it rises.
Spot the error
" where is the object's density."
Wrong density — buoyancy is the fluid pushing, so it must be . Object density only decides floating vs sinking, never the size of the upthrust.
"A steel ship floats because steel is lighter than water."
Steel is denser than water; the ship floats because its hull encloses air, dropping its average density below water's so it displaces enough water to balance its weight.
"The submerged fraction of a floating body is ."
Inverted — it is . With object less dense than fluid, as required; the wrong version would give a fraction bigger than 1.
"Since deeper fluid has higher pressure, buoyancy grows without limit as you sink."
For an incompressible fluid the difference across the object stays regardless of depth, so buoyancy is constant. (Only if the fluid compresses, like deep air, would it change.)
"The buoyant force acts at the object's centre of mass."
It acts at the centre of buoyancy = centroid of the displaced fluid volume, which coincides with the centre of mass only if the object is uniform. This distinction drives stability — see Floating bodies and stability — metacentre.
"A ball hanging on a string in water: the string tension equals the ball's full weight."
Tension equals (the apparent weight), because the water's upthrust already supports part of the weight.
"In the derivation the side faces contribute to the upward buoyant force."
Side faces come in opposite-facing pairs at equal depth, so their horizontal pressure forces cancel; only the top and bottom faces give the net vertical force.
Why questions
Why does vanish from the final buoyancy formula?
It appears identically in and , so it subtracts out; only the depth-difference term survives to become .
Why does the box shape used in the derivation not restrict the result to boxes?
The final carries no trace of "box"; replacing any shape by an identical blob of fluid (which floats in equilibrium) shows the surrounding fluid must push up with exactly that fluid's weight. See Newton's laws — equilibrium of forces.
Why is buoyancy directed upward and not sideways?
Pressure rises only with depth (vertically); horizontal pressures balance out, so the surviving imbalance is purely vertical, pointing from higher pressure (deep) toward lower pressure (shallow) — upward.
Why does an object feel "lighter" when weighed in water?
The water contributes an upward force , so the scale (supplying only the remainder) reads ; the mass hasn't changed, only the net supporting force needed.
Why does the same object float higher in seawater than in freshwater?
Seawater is denser ( larger), so a smaller submerged fraction suffices to balance the weight — less of the object sits underwater. See Density and relative density.
Why does buoyancy relate to Pascal's transmission of pressure?
Both stem from a fluid at rest transmitting pressure; here the depth-dependent part of that pressure field produces the net upthrust. See Pascal's principle.
Edge cases
An object resting flat on the tank bottom with no water sealed underneath: is there buoyancy?
No net upthrust — with no fluid pressing on the bottom face, the higher-pressure "up" force is missing, so the object can feel effectively glued down (like a suction cup) despite being submerged.
An object of density exactly equal to the fluid's, fully submerged and released: what happens?
It stays in neutral equilibrium — everywhere, so it neither rises nor sinks and hovers at whatever depth you leave it.
A sealed empty bottle held deep and released: does its buoyancy change as it rises?
For a rigid bottle, no — displaced volume is fixed, so is constant; a flexible bag would expand near the surface, increasing and thus buoyancy.
A completely weightless environment (free fall / orbit): is there buoyancy?
No — with , both hydrostatic pressure gradient and vanish; buoyancy needs gravity to create the depth-pressure difference.
An object floating with density equal to the fluid: what fraction is submerged?
, so it floats fully submerged with its top surface flush with the fluid — the boundary between floating and neutral suspension.
A dense object dropped in mercury (): why do steel and even lead float?
Their densities (steel , lead ) are below mercury's, so and they float — floating depends only on the density comparison, not on the object feeling "heavy."
Zero-volume limit: what buoyancy on an infinitely thin sheet lying horizontally?
With , — no displaced volume means no upthrust, consistent with there being no top-to-bottom depth difference to exploit.
Recall One-line takeaway to lock in
Buoyancy is the fluid's depth-pressure difference summed into a force pointing up; the object's own density only ever decides whether it floats, never how big the push is.
Connections
- Parent: Buoyancy topic note
- Pressure in fluids — hydrostatic pressure
- Density and relative density
- Apparent weight and weighing methods
- Floating bodies and stability — metacentre
- Pascal's principle
- Newton's laws — equilibrium of forces